Let be a rational function

(1)

where , is the Riemann sphere , and and are polynomials without common divisors. The "filled-in" Julia set is the set of points which do not approach infinity after is repeatedly applied (corresponding to a strange attractor). The true Julia set is the boundary of the filled-in set (the set of "exceptional points"). There are two types of Julia sets: connected sets (Fatou set) and Cantor sets (Fatou dust).

Quadratic Julia sets are generated by the quadratic mapping

(2)

for fixed . For almost every , this transformation generates a fractal. Examples are shown above for various values of . The resulting object is not a fractal for (Dufner et al. 1998, pp. 224-226) and (Dufner et al. 1998, pp. 125-126), although it does not seem to be known if these two are the only such exceptional values.

The special case of on the boundary of the Mandelbrot set is called a dendrite fractal (top left figure), is called Douady's rabbit fractal (top right figure), is called the San Marco fractal (bottom left figure), and is the Siegel disk fractal (bottom right figure).

The equation for the quadratic Julia set is a conformal mapping, so angles are preserved. Let be the Julia set, then leaves invariant. If a point is on , then all its iterations are on . The transformation has a two-valued inverse. If and is started at 0, then the map is equivalent to the logistic map. The set of all points for which is connected is known as the Mandelbrot set.

For a Julia set with , the capacity dimension is

(3)

For small , is also a Jordan curve, although its points are not computable.